The damped pendulum
When a damping force is added, that is to say that the pendulum is considered with real conditions, the stable equilibrium point becomes an attractive point and the unstable point becomes stable, as shown below.
Hereunder is a representative phase graph of a damped pendulum. We can see that all trajectories are attracted by the origin of the chart.
Moreover, as the drag coefficient becomes important, the trajectories tend to be asymptotically placed on an line. This is what we can see on the animation put below.